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Oscillation of linear Hamiltonian systems. (English) Zbl 1047.34030

The authors investigate oscillatory properties of the linear Hamiltonian system

${x}^{\text{'}}=A\left(t\right)x+B\left(t\right)u,\phantom{\rule{1.em}{0ex}}{u}^{\text{'}}=C\left(t\right)x-{A}^{T}\left(t\right)u,\phantom{\rule{2.em}{0ex}}\left(*\right)$

where $A,B,C$ are $n×n$-matrices with continuous entries for $t\in \left[{t}_{0},\infty \right)$, $B,C$ are symmetric and the matrix $B$ is supposed to be positive definite. Using a transformation which preserves the oscillatory nature of transformed systems, system (*) is transformed into the “diagonal off” system ${y}^{\text{'}}=\stackrel{˜}{B}\left(t\right)z$, ${z}^{\text{'}}=-\stackrel{˜}{C}\left(t\right)y$ with $\stackrel{˜}{B}$ positive definite, hence this system is equivalent to the second-order vector-matrix Sturm-Liouville differential equation

${\left({\stackrel{˜}{B}}^{-1}\left(t\right){y}^{\text{'}}\right)}^{\text{'}}+\stackrel{˜}{C}\left(t\right)y=0·\phantom{\rule{2.em}{0ex}}\left(**\right)$

The main result of the paper is formulated under the assumption that the matrix $\stackrel{˜}{B}\left(t\right)-I$ is nonnegative definite ($I$ denotes the identity matrix). Under this assumption, system (**) is a Sturmian majorant of the system ${y}^{\text{'}\text{'}}+\stackrel{˜}{C}\left(t\right)y=0$. So, oscillation of the last system implies oscillation of (**) and hence, in turn, also of (*). From this point of view, the results of the paper are very close to those presented in the paper of G. J. Butler, L. H. Erbe and A. B. Mingarelli [Trans. Am. Math. Soc. 303, 263–282 (1987; Zbl 0648.34031)].

MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34A30 Linear ODE and systems, general